Pseudo-R-squared

R²_L is given by Cohen:^[1]

${\displaystyle R_{\text{L}}^{2}={\frac {D_{\text{null}}-D_{\text{fitted}}}{D_{\text{null}}}}.}$ 

This is the most analogous index to the squared multiple correlations in linear regression.^[3] It represents the proportional reduction in the deviance wherein the deviance is treated as a measure of variation analogous but not identical to the variance in linear regression analysis.^[3] One limitation of the likelihood ratio R² is that it is not monotonically related to the odds ratio,^[1] meaning that it does not necessarily increase as the odds ratio increases and does not necessarily decrease as the odds ratio decreases.

R²_CS is an alternative index of goodness of fit related to the R² value from linear regression.^[2] It is given by:

${\displaystyle {\begin{aligned}R_{\text{CS}}^{2}&=1-\left({\frac {L_{0}}{L_{M}}}\right)^{2/n}\\[5pt]&=1-\exp \left({\frac {2}{n}}(\ln(L_{0})-\ln(L_{M}))\right)\end{aligned}}}$ 

where L_M and L₀ are the likelihoods for the model being fitted and the null model, respectively. The Cox and Snell index corresponds to the standard R² in case of a linear model with normal error. In certain situations, R²_CS may be problematic as its maximum value is ${\displaystyle 1-L_{0}^{2/n}}$ . For example, for logistic regression, the upper bound is ${\displaystyle R_{\text{CS}}^{2}\leq 0.75}$  for a symmetric marginal distribution of events and decreases further for an asymmetric distribution of events.^[2]

R²_N, proposed by Nico Nagelkerke in a highly cited Biometrika paper,^[4] provides a correction to the Cox and Snell R² so that the maximum value is equal to 1. Nevertheless, the Cox and Snell and likelihood ratio R²s show greater agreement with each other than either does with the Nagelkerke R².^[1] Of course, this might not be the case for values exceeding 0.75 as the Cox and Snell index is capped at this value. The likelihood ratio R² is often preferred to the alternatives as it is most analogous to R² in linear regression, is independent of the base rate (both Cox and Snell and Nagelkerke R²s increase as the proportion of cases increase from 0 to 0.5) and varies between 0 and 1.

The pseudo R² by McFadden (sometimes called likelihood ratio index^[5]) is defined as

${\displaystyle R_{\text{McF}}^{2}=1-{\frac {\ln(L_{M})}{\ln(L_{0})}},}$ 

and is preferred over R²_CS by Allison.^[2] The two expressions R²_McF and R²_CS are then related respectively by,

${\displaystyle {\begin{matrix}R_{\text{CS}}^{2}=1-\left({\dfrac {1}{L_{0}}}\right)^{\frac {2(R_{\text{McF}}^{2})}{n}}\\[1.5em]R_{\text{McF}}^{2}=-{\dfrac {n}{2}}\cdot {\dfrac {\ln(1-R_{\text{CS}}^{2})}{\ln L_{0}}}\end{matrix}}}$ 

Allison^[2] prefers R²_T which is a relatively new measure developed by Tjur.^[6] It can be calculated in two steps:

1. For each level of the dependent variable, find the mean of the predicted probabilities of an event.
2. Take the absolute value of the difference between these means

A word of caution is in order when interpreting pseudo-R² statistics. The reason these indices of fit are referred to as pseudo R² is that they do not represent the proportionate reduction in error as the R² in linear regression does.^[1] Linear regression assumes homoscedasticity, that the error variance is the same for all values of the criterion. Logistic regression will always be heteroscedastic – the error variances differ for each value of the predicted score. For each value of the predicted score there would be a different value of the proportionate reduction in error. Therefore, it is inappropriate to think of R² as a proportionate reduction in error in a universal sense in logistic regression.^[1]